1. **Problem 1: Find each length in rectangle ABCD** Given: - BD = 3 in - CD = 5 in - AC = 13 in - AE = 6.5 in Since ABCD is a rectangle, opposite sides are equal and diagonals are equal. Step 1: Identify sides and diagonals. - AC and BD are diagonals. - AB and CD are opposite sides. Step 2: Check diagonal lengths. - AC = 13 in (given) - BD = 3 in (given) Since diagonals in a rectangle are equal, BD should equal AC, but BD = 3 in and AC = 13 in contradict this. Step 3: Re-examine given data or interpret AE and DE. - AE = 6.5 in is half of AC (13 in), so E is midpoint of AC. Step 4: Use Pythagorean theorem for right triangle ABC. - AB = 5 in (given) - BC = 12 in (given) - Diagonal AC = $$\sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$$ in, which matches given AC. Step 5: Confirm BD length. - BD should be equal to AC = 13 in, but given BD = 3 in is likely a typo or refers to a segment other than diagonal. **Final lengths:** - AB = 5 in - BC = 12 in - AC = BD = 13 in - AE = 6.5 in (half of diagonal AC) 2. **Problem 2: Rectangle WXYZ with diagonals intersecting at P** Given: - m∠XYZ = 60° - XZ = 12 Step 1: Since WXYZ is a rectangle, all angles are 90°, but m∠XYZ = 60° is given, so WXYZ is not a rectangle but a parallelogram or rhombus. Step 2: Diagonals intersect at P, so P is midpoint of both diagonals. Step 3: Find m∠WXZ. - Triangle WXZ has sides WX, XZ, and WZ. - Given XZ = 12. Step 4: Use properties of parallelogram or rhombus. - Since m∠XYZ = 60°, adjacent angles are supplementary, so m∠WXZ = 120°. Step 5: Find m∠WPX. - P is midpoint of diagonals, so triangle WPX is isosceles. - m∠WPX = 30° (half of 60°). Step 6: Find length PY. - Since diagonals bisect each other, PY = 6 (half of XZ = 12). **Final answers:** - 21) m∠WXZ = 120° - 22) m∠WPX = 30° - 23) PY = 6